Question

solve the following system using the elimination method. enter your answer as an ordered pair in the form ((x, y)) if there is one, unique solution. enter all if there are infinitely many solutions and enter none if there are no solutions.
(6x - 8y = 14)
(3x + 2y = 37)

Explanation:

Step1: Multiply the second equation by 4 to eliminate y

We have the system:

$$\begin{cases} 6x - 8y = 14 \\ 3x + 2y = 37 \end{cases}$$

Multiply the second equation \(3x + 2y = 37\) by 4, we get:
\(4\times(3x + 2y)=4\times37\)
\(12x + 8y = 148\)

Step2: Add the two equations to eliminate y

Now we have the first equation \(6x - 8y = 14\) and the new second equation \(12x + 8y = 148\). Add them together:
\((6x - 8y)+(12x + 8y)=14 + 148\)
\(6x+12x-8y + 8y=162\)
\(18x=162\)

Step3: Solve for x

Divide both sides of \(18x = 162\) by 18:
\(x=\frac{162}{18}\)
\(x = 9\)

Step4: Substitute x = 9 into the second original equation to solve for y

Substitute \(x = 9\) into \(3x+2y = 37\):
\(3\times9+2y=37\)
\(27+2y=37\)
Subtract 27 from both sides:
\(2y=37 - 27\)
\(2y=10\)
Divide both sides by 2:
\(y=\frac{10}{2}\)
\(y = 5\)

Answer:

\((9,5)\)